where \(\Psi ^*_\mathcal {A}(t,a) = * \circ \Psi _\mathcal {A}(t,a) = \Psi _\mathcal {A}(-t,a)\)  by Eq. (REF ). In particular, \(\Phi _\rho \)  in \(\phi _\rho = v^*\Phi _\rho v\)  is a \(C^*\) -homomorphism [1]}. If \(\rho \)  is also time-oriented with respect to \(\mathcal {A}_-\)  and \(\mathcal {B}_-\) , then by Def. REF ,
\(\begin{split}&\forall t \in \mathbb {R}, a \in \mathcal {J}(\mathcal {A})_\mathrm {sa}:\ \Phi _\rho \circ \Psi ^*_\mathcal {A}(t,a) = \Psi ^{\prime *}_\mathcal {B}(t,\Phi (a)) \circ \Phi _\rho \\\Longleftrightarrow \quad &\forall t \in \mathbb {R}, a \in \mathcal {J}(\mathcal {A})_\mathrm {sa}:\ \Phi _\rho \circ \Psi _\mathcal {A}(t,a) = \Psi ^{\prime }_\mathcal {B}(t,\Phi (a)) \circ \Phi _\rho \; .\end{split}\) 
